SERIES. 
» ^ 
Theorem 5. In the arithmetic progression 
1, 2, 3, 4, &c. the sum is to the product of 
the last term, by the number of terms, 
that is, to the square of the last terra ; in a 
ratio always greater than 1 : 2, but ap- 
proaching infinitely near it. But if the 
arithmetical series begins with 0, thus, 0, 1, 
2, 3, 4, &c. then the sum is to the product 
of the last term, by the number of terms, 
exactly in every step as 1 to 2. 
Theorem 6. Take the natural progression 
beginning with 0, thus, 0, 1, 2, 3, &c. and 
take the squares of any the like powers of 
the former series, as the squares, 0, 1, 4, 9, 
&c. or cubes, 0, 1, 8, 27 ; and then again 
take tlie sum of the series of powers to any 
number of terms, and also multiply the last 
of the terms summed by the number of 
terms, (reckoning always 0 for the first 
term) the ratio of that sum, to that product 
is more than 
(n being the index of tlie 
m X 1 
powers), that is, in the series of squares it 
is more than 
in the cubes more than ■ 
and so on ; but the series going on in infini- 
tum, we may take in more and more terms 
without end into the sura; and the more we 
take, the ratio of the sura to the product 
mentioned grows less and less ; yet so as it 
never can actually be equal to ^ ^ ^ but 
approaches infinitely near to it, or within 
less than any assignable difference. 
“ The nature, origin, &c. of series.” In- 
finite series commonly arise, either from a 
continued division, or the extraction of root?, 
as first performed by Sir I. Newton, who 
also explained other general ways for the 
expanding of quantities into infinite series, 
as by the binomial theorem. Thus, to di- 
vide 1 by 3, or to expand the fraction r into 
an infinite series; by division in decimals in 
the ordinary way, the series is 0.3333, &c. or 
3,3, 3 
_ J 1:L -L — - -L _ 
10 ‘ 100 ‘ 1000 ‘ lOOOO' 
&c. where the 
law of continuation is manifest. Or, if the same 
fraction \ be set in this form — ^ — , and di- 
3 2-t-l’ 
vision be performed in the algebraic man- 
ner, the quotient will be 
1 1 _1 
3 “a-l-i — a 
.1 + 1 . 
4~8 
1 . f . 
Or, if it be expressed in this form ~ : 
4— t’ 
the series. 
by a like division there will arise 
1 — 1+i+J- &c — l-l-i-4-i 
4-7- 421-43' 
&c. 
And, thus, by dividing 1 by 5 — 2, or 6 — 3, 
or 7 — 4, &c. the series answering to the 
fraction i, may be found in an endless variety 
of infinite series ; and the finite quantity i is 
called the value or radix of the series, or 
also its sura, being the number or sum to 
which the series would amount, or the limit 
to which it would tend or approximate, by 
summing up its terms, or by collecting them 
together one after another. In like man- 
